Integration of Hamiltonian systems of hydrodynamic type with two dependent variables with the aid of the Lie-Bäcklund group
Many equations, admitting the Lax operator representation, are the continual analogues of Hamiltonian systems. A method of construction of infinite series of commuting Hamiltonians with the aid of generating functions has been indicated by \textit{Yu. I. Manin} [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 11, 5-152 (1978; Zbl 0413.35001)]. We construct a matrix integral operator K, whose powers generate the entire series of the Manin Hamiltonians. The operator K forms an L-A pair together with another operator, defined directly from the Hamilton equations. The inverse operator \(K^{-1}\) is connected to the gauge transformation with the matrix differential operator L, which is the recurrence operator of the one-parameter Lie-Bäcklund groups, admitted by the given system of equations. The operator L, together with the operator of the defining equation of the admitted group, forms an ``L-A pair of the Ibragimov-Shabat type''. With the aid of such an L-A pair one computes the infinite nontrivial Lie-Bäcklund group, connected with a special class of Hamiltonian systems of hydrodynamic type with two dependent variables. With the aid of the operator \(L^{-1}\) one obtains infinite series of invariant solutions. Thus, the L-A pair of the Ibragimov-Shabat type allows us to compute the symmetries and the exact solutions of a nonlinear system of equations, even in the case when it is degenerate (the mapping of the potential into the scattering data is singular) and the method of the inverse scattering problem cannot be applied. The particularity of the method presented in this paper is the fact that for the construction of such a L-A pair one does not have to solve the Lax equation or the defining equation of the admitted group. We also show that the formulas for the solutions, invariant relative to the Lie-Bäcklund groups with a functional arbitrariness, represent a generalization of the hodograph transformation to the case of a system with an explicit time dependence. The problem of the integration of a system of hydrodynamic type and the determination for them of an analogue of the inverse problem method has been formulated by \textit{B. A. Dubrovin} and \textit{S. P. Novikov} [Dokl. Akad. Nauk. SSSR 270, 781-785 (1983; Zbl 0553.35011)]. The results of the present paper give the solution of this problem in one simple case.
construction of infinite series of commuting Hamiltonians
scientific article; zbMATH DE number 4035372